关于初始数据无界集合上随机热方程的一致大偏差原理
Uniform large deviation principles for the stochastic heat equation over unbounded sets of initial data
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中文总结 AI 辅助
该研究针对环面上带无界乘性时空白噪声的随机热方程，建立了初始数据无界集合上的一致大偏差原理，将其适用范围拓展至$L^1$有界子集，可用于研究质量守恒物理系统的首出时问题。
中文摘要 AI 辅助
我们研究环面上带无界乘性时空白噪声的随机热方程（SHE）的小噪声大偏差，建立了初始数据无界集合上的一致大偏差原理（ULDPs），同时得到了新的适定性与正则性结果。ULDPs针对空间连续和$L^p$初始数据两类情形，分别在$L^q$有界子集上一致成立，允许$q<\text{infty}$且$q<p$，这类集合在状态空间中高度无界。$q$的允许范围由SHE中噪声系数的增长性决定，关键是我们的方法能达到$q=1$，这使得$L^1$有界子集上的ULDPs得以建立，可用于研究质量守恒的物理系统中的首出时问题。
英文摘要
We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and $L^p$ initial data. For these two classes, the ULDP holds uniformly over $L^q$-bounded subsets, with $q<\infty$ and $q<p$ allowed respectively, so these sets are highly unbounded in the state space. The admissible range of $q$ is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to $q=1$. This yields ULDPs over $L^1$-bounded subsets, enabling the study of exit times in physical systems with mass conservation.
